On locally compact shift-continuous topologies on semigroups $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ with adjoined zero

TL;DR

This paper investigates the variety of Hausdorff locally compact shift-continuous topologies that can be imposed on certain semigroups with adjoined zero, revealing a rich diversity of such topologies.

Contribution

It demonstrates that the semigroups with adjoined zero admit continuum many distinct Hausdorff locally compact shift-continuous topologies up to isomorphism.

Findings

Existence of continuum many topologies

Topologies are Hausdorff and locally compact

Topologies are shift-continuous

Abstract

Let $\mathscr{C}_{+}(p,q)^0$ and $\mathscr{C}_{-}(p,q)^0$ be the semigroups $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ with the adjoined zero. We show that the semigroups $\mathscr{C}_{+}(p,q)^0$ and $\mathscr{C}_{-}(p,q)^0$ admit continuum many different Hausdorff locally compact shift-continuous topologies up to topological isomorphism.